In the 80's, Crandall and Lions introduced the concept of viscosity solution, in order to get existence and/or uniqueness results for Hamilton-Jacobi equations. In this work, we first investigate the Dirichlet and Cauchy-Dirichlet problems for such equations, where the Hamiltonian is associated to a problem of calculus of variations, and prove that, if the data are analytic, then the viscosity solution is moreover subanalytic. We then extend this result to Hamilton-Jacobi equations stemming from optimal control problems, in particular from sub-Riemannian geometry, which are generalized eikonal equations. As a consequence, the set of singularities of the viscosity solutions of such Hamilton-Jacobi equations is a subanalytic stratified manifold of codimension greater than or equal to one.
